Notation Used for Arc Length

The angle $\,\theta\,$ can be in degrees or radians,
providing you use the correct formula. Keep reading!

The units for $\,r\,$ and $\,s\,$ are (the same) length unit: e.g., feet, meters, miles, centimeters.

For example, if $\,r\,$ is given in (say) feet, then a computed value for $\,s\,$ will also be feet.
Or, if $\,s\,$ is given in (say) centimeters, then a computed value for $\,r\,$ will also be centimeters.

Length of a Circular Arc: Using Degree Measure

In degree measure, one complete revolution is $\,360^\circ\,$.

The circumference of a circle with radius $\,r\,$ is $\,2\pi r\,$.
Circumference takes the unit of $\,r\,$, since the
real number $\,2\pi\,$ has no units.

Let $\,\theta\,$ be a real number, without units,
that represents the degree measure of a central angle.
For example, if $\,\theta = 60\,$, then $\,\theta^\circ = 60^\circ\,$.

Both ways will work, but they give rise to slightly-different-looking formulas.

At left, choice (2) was used, yielding the formula:
$$
s = \frac{\pi r\theta}{180}
$$
For choice (1), the degree units can't be cancelled, since the unit is ‘trapped’
inside the variable $\,\theta\,$.
In this case, the formula becomes:
$$
s = \frac{\pi r\theta}{180^\circ}
$$
Of course, when you put

Subsitution of this radian measure into the formula $\,s = r\theta\,$ gives
$$
s \ =\ r\left(\frac{\pi\theta}{180}\right) \ =\ \frac{\pi r\theta}{180}
$$
which, of course, is the same formula derived above. Thus, we see that the
‘$\frac{\pi}{180}\,$’ is just a conversion factor from degree to radian measure!

Why the Name ‘Radian Measure’?

The formula $\,s = r\theta\,$ gives a beautiful insight into why the
name ‘radian’ is used for radian measure.
The idea was discussed in an earlier section on
radian measure, but is
worth repeating here!